MathPlotAI Logo MathPlotAI

Dynamic Relationship Between Triangle's Altitudes, Circumcircle, and Tangent

Dynamic Relationship Between Triangle's Altitudes, Circumcircle, and Tangent
Loading the math board and drawing, please wait…
2017-05-23 08:49
TriangleCircleGeometric ConstructionLocus
This example demonstrates the dynamic and modifiable relationship between two altitudes of a triangle, its circumcircle, and a tangent drawn through its third vertex. Key math concepts include circumcircle, multiplication, division, altitudes, tangents, rotation, vectors, and more. Construction highlights GeoGebra commands like Rotate, Ray, Point, Segment, Line, OrthogonalLine, Intersect, Min, Circle, Angle, LineBisector, CircleArc, Dilate, CircleSector, Vector, Translate, and Polygon. Suitable for high school and above students exploring geometric relationships and dynamic geometry software applications.

GeoGebra Commands & Parameters

  1. Rotate(B, (a * π / 180), A) → B'
  2. Ray(A, B') → g
  3. Point(g) → C
  4. Segment(A, C) → q_1
  5. Line(C, B) → r
  6. Line(A, C) → q
  7. OrthogonalLine(B, q) → f_1
  8. Intersect(q, f_1) → H
  9. Segment(C, H) → r_1
  10. Min(q_1, r_1) → v
  11. Circle(C, (0.4 * v)) → s_1
  12. Intersect(s_1, r) → O, P
  13. Intersect(s_1, q) → M, N
  14. Angle(M, C, P) → ζ
  15. Segment(C, B) → j
  16. LineBisector(j) → k
  17. Segment(A, C) → i
  18. LineBisector(i) → l
  19. Intersect(k, l) → D
  20. Line(D, C) → m
  21. OrthogonalLine(C, m) → n
  22. Intersect(s_1, n) → Q, R
  23. Angle(A, C, Q) → κ
  24. Angle(B, A, B') → α
  25. Ray(A, B) → f
  26. Segment(A, B) → h
  27. Rotate(C, If[0 < b ≤ 2, ((b - 2) * π), (2 * π)], D) → C'
  28. CircleArc(D, C, C') → c
  29. Circle(D, C) → d
  30. Circle(C, 3) → e
  31. Intersect(e, n, 1) → E
  32. Intersect(e, n, 2) → F
  33. Angle(D, C, E) → β
  34. Dilate(D, If[2 < b ≤ 3, 3 - b, 0], C) → D'
  35. Segment(D, D') → p
  36. OrthogonalLine(A, r) → s
  37. Intersect(r, s) → G
  38. Angle(A, G, B) → γ
  39. Dilate(A, If[3 < b ≤ 4, 4 - b, 0], G) → A'
  40. Segment(A, A') → t
  41. Dilate(B, If[4 < b ≤ 5, 5 - b, 0], H) → B'_1
  42. Segment(B, B'_1) → g_1
  43. Angle(A, H, B) → δ
  44. Dilate(H, If[5 < b ≤ 6, 6 - b, 0], G) → H'
  45. Segment(H, H') → h_1
  46. Line(H, G) → i_1
  47. Segment(H, C) → j_1
  48. Segment(H, A) → k_1
  49. Intersect(d, i_1) → I, J
  50. Segment(H, G) → l_1
  51. Segment(H, I) → m_1
  52. Min({j_1, k_1, l_1, m_1}) → o
  53. Circle(H, (0.4 * o)) → c_1
  54. Intersect(c_1, q, 1) → K
  55. Intersect(c_1, i_1, 2) → L
  56. Angle(K, H, L) → ε
  57. Rotate(K, If[7 < b ≤ 8, ((b - 7) * ε), ε], H) → K'
  58. CircleSector(H, K, K') → d_1
  59. Dilate(H, If[8.5 < b ≤ 9.5, 9.5 - b, 0], C) → H'_1
  60. Vector(H, H'_1) → u
  61. Translate(d_1, u) → e_1
  62. Polygon(A, B, C) → poly1, c_2, a_1, b_1
  63. Segment(A, B) → n_1
  64. Segment(A, H) → p_1
  65. Rotate(M, If[7 < b ≤ 8, ((b - 7) * κ), κ], C) → M'
  66. CircleSector(C, M, M') → t_1
  67. Dilate(C, If[8.5 < b ≤ 9.5, 9.5 - b, 0], H) → C'_1
  68. Vector(C, C'_1) → w
  69. Translate(t_1, w) → d_2
  70. Dilate(D, If[-1 < b < 0, (-b), 0], C) → D'_1
  71. Segment(D, D'_1) → f_2
  72. Rotate(D, If[0 < b ≤ 2, ((b - 2) * π), (2 * π)], D) → D'_2
  73. Rotate(D'_1, If[0 < b ≤ 2, ((b - 2) * π), (2 * π)], D) → T
  74. Segment(D'_2, T) → g_2
  75. Segment(C, A) → h_2
  76. Min(h_2, n_1) → o_1
  77. Circle(A, (0.15 * o_1)) → e_2
  78. Intersect(e_2, n_1) → U,
  79. Intersect(e_2, h_2) → V,
  80. CircleSector(A, U, V) → k_2